General Equilibrium Problem Set Flashcards
- a) Exercise 1 There are two utility maximizing consumers i ∈ {1, 2} in a country, and there
exist two goods, x and y. The utility functions of the consumer 1 and consumer 2 are given by
U1(X1, Y1) = X1 ^ 1/4 * Y1^3/4
U2(X2, Y2) = X2* Y2^1/2
respectively. The initial endowments of the
two consumers are given by ω1 = (2, 0) and ω2 = (0, 1) respectively.
Find the competitive equilibrium. What are the quantities that are consumed by
consumer 1 and consumer 2 in the competitive equilibrium?
The Competitive Equilibrium has two conditions
- Consumers should maximize Utility function subject to the constraint
- Market should clear, meaning market supply = market demand
Given:
U1(X1, Y1) = X1 ^ 1/4 * Y1^3/4
ω1 = (2, 0)
U2(X2, Y2) = X2* Y2^1/2
ω2 = (0, 1)
- Set up the budget constraint
Budget Constraint for Consumer 1:
Px1 + y1 = Pwa^x + wa^y
Px1+ y1 = 2p
Budget Constraint for Consumer 2:
Px2 + y2 = Pwb^x + wb^y
Px2 + y2 = 1
- Is use the maximizing condition that MRS = PX/PY knowing that PX/PY = P
Consumer 1:
MRS = PX/PY
MUX/MUY = P/1
1/4x1^-3/4*y1^3/4 / 3/4y1^-1/4x1^1/4 = P/1
Solve for Y1
Y1 = 3x1P
Consumer 2:
MRS = PX/PY
MUx/MUY = P/1
Y2^1/2 / 1/2y2^-0.5x2 = P/1
Solve for Y2
Y2 = Px2/ 2
- Use the budget constraints and the intermediate demands to solve for the demand equations
Consumer 1:
Budget Constraint: 2P = Px1 + y1
Y1 = 3x1P, sub in
2P = Px1 + y1
2P = Px1 + 3x1P
Solve for X1
x1 = 1/2
Sub x1 = 1/2 back into Y1 = 3x1P
Y1 = 31/2P = 3/2P
Consumer 2:
Budget Constraint:
Px2 + y2 = 1
Plug Y2 = Px2/ 2
Px2 + Px2/ 2 = 1
Solve for x2
x2 = 2/3 * 1/P
- Now that we have x1 and x2 solve for p by setting the clearing condition for the market of good x
x1 + x2 = 2
1/2 + 2/3 * 1/P = 2
Solve for P
P = 4/9
- Plug all the values in to find the quantities
x1 = 1/2
y1 = 3/2p = 3/2 (4/9) = 2/3
x2= 2/3(1/P) = 2/3(1/4/9) = 3/2
y2 = Px2/2 = 4/9*(3/2) / 2 = 1/3
- Now checking if P clears the market for y
y1 + y2 = 1
2/3 + 1/3 = 1
1 = 1
The market for y clears - Use the Utility functions to calculate the utility at these quantities.
U1(X1, Y1) = U1(1/2, 2/3) = 0.62
U2(X2, Y2) = U2(3/2, 1/3) = 0.866
- b) Suppose that the country signs a free-trade agreement with consumer 3. The utility
function of consumer 3 is given by
U3(X3, Y3) = X^1/2 * Y3
and their endowment is given
by ω3 = (0, 1). Is consumer 1 better off? Is consumer 2?
From Part A:
U1(X1, Y1) = U1(1/2, 2/3) = 0.62
U2(X2, Y2) = U2(3/2, 1/3) = 0.866
x1 = 1/2
y1 = 3/2p = 3/2 (4/9) = 2/3
x2= 2/3(1/P) = 2/3(1/4/9) = 3/2
y2 = Px2/2 = 4/9*(3/2) / 2 = 1/3
- Write the Budget Constraint:
Px3 + y3 = Pwa^x + wb^y
Px3 + y3 = 1
- Maximize Consumer 3’s Utility
MRS = PX/PY
MUX/MUY = P/1
1/2X3^-0.5Y3 / X3^1/2 = P/1
Solve for Y3
Y3 = 2PX3
- Sub Y3 = 2PX3 into the constraint
Px3 + y3 = 1
Px3 + 2PX3 = 1
Solve for X3
X3 = 1/3P
- Check that the market for X clears and solve for P
X1 + X2 + X3 = 2
1/2 + 2/3 (1/P) + 1/3P = 2
P = 2/3
- Find all the values again:
x1 = 1/2
y1 = 3/2(2/3) = 1
x2= 2/3(1/P) = 2/3(1/2/3) = 1
y2 = Px2/2 = 2/3(1)/2 = 1/3
x3 = 1/3P = 1/3(2/3) = 1/2
Y3 = 2PX3 = 2(2/3)(1/2) = 2/3
- Find the utility to see who is better off using the functions
U1(X1,Y1) = U1(1/2,1) = 0.84
U2(X2,Y2) = U2(1, 1/3) = 0.577
Consumer 1 who supplies relatively more scarce x is going to benefit whereas consumer 2 who sells y is worse off.
- a) In a country, there are two consumers, 1 and 2, and two goods, x and y. The
consumers have the following utility functions and endowments:
U1(X1, Y1) = X1^1/2 *Y1^1/2
w1 = ( 2,6)
U2(X2, Y2) = X2^1/4 * Y2^3/4
w2 = (8,2)
(a) Determine the competitive equilibrium.
- Find the budget constraints
Consumer 1:
PX1 + Y1 = Pwa^x + wb^y
PX1 + Y1 = 2P + 6
Consumer 2:
PX2 + Y2 = Pwa^x + wb^y
PX2 + Y2 = 8P + 2
- Maximize utility using the condition that MRS = PX / PY for each consumer
Consumer 1:
MRS = PX/ PY
MUX/MUY = P/1
1/2X1^-1/2Y1^1/2 / X1^1/2*1/2Y1^-1/2 = P/1
Solve for Y1
Y1 = Px1
Consumer 2:
MRS = PX/ PY
MUX/MUY = P/1
1/4X2^-3/4 *Y2^3/4 / X2^1/4 *3/4Y2^-1/4 = P/1
Solve for Y2
Y2 = 3PX2
3.Use the budget constraints and the intermediate demands to solve for the demand equations.
Use Consumer 1’s Budget Constraint:
Px1 + y1 = 2p + 6
Sub Y1 = Px1
Px1 + Px1 = 2p + 6
Solve for X1
x1 = 1 + 3/p
Plug back to Y1
Sub Y1 = Px1
Y1 = Px1
Y1 = P(1+ 3/P)
Use Consumer 2’s Budget Constraint:
PX2 + Y2 = 8P + 2
Sub Y2 = 3PX2
PX2 + Y2 = 8P + 2
PX2 + 3PX2 = 8P + 2
Solve for X2
X2 = 2 + 1/2P
- Check if the market clears for X
X1 + X2 = 10
1 + 3/P + 2 + 1/2P = 10
Solve for P
P = 7/14 = 1/2
- Find all the values:
X1 = 1 + 3/P = 1 + 3/1/2 = 7
X2 = 2 + 1/2P = 2+ 1/2(1/2) = 3
Y1 = P (1+ 3/P) =1/2(1 + 3/1/2) = 7/2
Y2 = 3PX2 = 3(1/2)(3) = 9/2
- Check if the Y- market clears
Y1 + Y2 = 8
7/2 + 9/2 = 8
16/2 = 8
8 = 8, the market for Y clears - Find the Utility’s
U1(X1, Y1) = U1(7, 7/2) = 4.95
U2(X2,Y2) = U2(3, 9/2) = 4.066
2.b) Assume that the country concludes a free trade agreement with consumer 3, such that there is free trade between 1, 2 and 3. The endowments and preferences of consumer
3 are:
U3(X3, Y3) = X3^1/4Y^3/4
w3 = (8,2)
What is the competitive equilibrium under free trade? Is the free trade agreement a
Pareto improvement?
From part 2a)
U1(X1, Y1) = U1(7, 7/2) = 4.95
U2(X2,Y2) = U2(3, 9/2) = 4.066
X1 = 1 + 3/P
X2 = 2 + 1/2P
Y1 = P (1+ 3/P)
Y2 = 3PX2
- Write the Budget Constraint
PX3 + Y3 = Pwa^x + wb^y
PX3 + Y3 = 8P + 2
Consumer 3 has the same utility function in terms of preferences and endowments
X3 = 2 + 1/2P
Then Y3 = 3PX3
Y3 = 3PX3
Y3 = 3P( 2 + 1/2P)
- Check that the market for X clears and solve for X
X1 + X2 + X3 = 18
1 + 3/P + 2 + 1/2P + 2 + 1/2P = 18
Solve for P
P = 4/13
- Now Solve for the values where P = 4/13 sub this in.
X1 = 1 + 3/P = 43/4
X2 = 2 + 1/2P = 29/8
X3 = 2 + 1/2P = 29/8
Y1 = P (1+ 3/P) = 43/13
Y2 = 3PX2 = 87/26
Y3 = 3PX3 = 87/26
U1(X1, Y1) = U1(43/4, 43/13) = 5.96
U2(X2,Y2) = U2(29/8, 87/26) = 3.41
U3(X3, Y3) = U3(29/8, 87/26) = 3.41
Consumer 1 is better off 5.94>4.95
Consumer 2 is worse off 3.41 < 4.06
Consumer 3:
Utility of endowment
U3(X3, Y3) = X3^1/4Y^3/4
U3(X3, Y3) = (8)^1/4(2)^3/4 = 2.82
Consumer 3 is better off since her utility of consuming her endowment is 2.82 and
3.41 > 2.82
General Equilibrium 1: Problem
Refer to General Equilibrium Problem Set Diagram
What type of goods are x and y for consumer 1 and consumer 2 respectively? Find a utility function that represents the preferences of consumer 1.
The two goods x and y are perfect complements for consumer 1 and imperfect substitutes for consumer 2.
Consumer 1’s Utility function is:
It represent the ratio 1: 2 which is the same as 1/2x = Y
X Y
1. 1/2
2. 1
3. 3/2
4. 2
U1(x1, y1) = min[2x1, y1]
General Equilibrium 1: Problem
Refer to General Equilibrium Problem Set Diagram
Does consumer 2 obtain the same utility from allocations B and C? How about A and
C? Does consumer 1 get the obtain the same utility from allocations (1, 2) and C?
How about (1, 2) and A?
Consumer 2 is indifferent between allocations A, B, C since they lie on the same indifference curve.
Consumer 1 is indifferent between (1,2) and C, however strictly prefers A over (1,2)
General Equilibrium 1: Problem
Refer to General Equilibrium Problem Set Diagram
Is C Pareto efficient? Is (1, 2)? Find the contract curve
In order to be pareto efficient it is such a case where it is impossible to make one person better off without making the other person worse off.
With point C, it is not Pareto efficient, consumer one would remain on the same indifference cure, while consumer 2 would obtain a higher indifference curve, this is a Pareto improvement.
Point (1, 2) is pareto efficient. It is not possible to make one person better off without making the other worse off. In other words, its not possible to increase the utility of one consumer without decreasing the utility of the other consumer.
If you remove one unit of x and give it to consumer 2, consumer 2 would be worse off.
The contract curve shows the set of all pareto allocations, and ensures consumer 1 at least gets on indifference curve 1, and consumer 2 at least gets on indifference curve 2
The contract curve is y1(x1) = 2x1
General Equilibrium 1: Problem
Refer to General Equilibrium Problem Set Diagram
Consider the Edgeworth Box.
A = (XA, Y A) and B = (XB, Y B) and C = (XC, Y C). Allocation ω denotes the endowment
Suppose that, at P, consumer 1 chooses to consume (XA, Y A) and consumer 2 chooses
to consume (X¯ − XA, Y¯ − Y)
A). Does the market for x clear? Does the market for y
clear? Is (XA, Y A) Pareto effcient?
Y bar and X bar represent the amount of good X bar and Y bar is in the market.
In the question we know for good x consumer one chooses X^A and for consumer two it chooses
X bar - X^A. The market clearing condition is satisfied
X1 + X2 = X^A + (Xbar - X^A)
= X^A + Xbar - X^A
= X bar
By Walras law, stating the value of aggregate excess demand is zero for all prices. Since the market for X clears the market for Y also clears.
Y1 + Y2 = Y^A + (Ybar - Y^A)
= Y^A + Ybar - Y^A
= Ybar
The initial endowment of both consumers is w. Consumer 1 started at w1.
The supply of x by consumer 1 is greater than the demand of x by consumer 2. This means at these prices there is an excess supply of consumer 1 at prices (Px, Py). If there is an excess supply of x, the price of x should decrease px down.
The demand of y by consumer 1 is greater than the supply of y by consumer 2. This means at these prices theres excess demand of consumer 1 at prices (Px, Py). If there is excess demand of something the price of y should increase Py goes up.
Initially in the BL we had px/py now we have px’/py’. Since px is decreasing and py is increasing than px’/py’ falls, going through the endowment point. This price change process will continue until equilibrium is reached.
Equilibrium is reached when both maximize at the same pint, or when all gains from trade are exhausted. At this point, there is two tangency conditions
MRSxy1 = PY/PY and
MRSxy2 = PX/PY, which is written as
MRSXY^A = MRSXY^B = PX/PY.
This is the condition the competitive equilibrium point is also Pareto efficient.
General Equilibrium 1: Problem
Refer to General Equilibrium Problem Set Diagram
Suppose that, after the relative price has changed from P to P ′, consumer 1 chooses
consumption bundle B and consumer 2 chooses consumption bundle C. Does good x become more or less expensive than good y? Is the market in equilibrium at P ′?
At competitive equilibrium :
- Excess supply of good x of consumer 1 > Excess demand of good x by consumer 2
Xb + (Xbar - Xc) < X bar
we can see this condition is not met
- Excess demand of good y of consumer 1 > Excess Supply of good y of consumer 2
Yb + (Ybar - Yc) > Ybar
From consumption, we know that the budget line is
BL(P) = P(w1^x - X1) + w1^y
Originally we had px/py now we have px’/py’, this shifted up which means excess supply should occur at equilibrium, so px decreases, instead there is excess demand so px goes up. At competitive equilibirum, Py increases with excess demand in good x, instead there excess supply so Px increases.
General Equilibrium 1: Problem
Refer to General Equilibrium Problem Set Diagram
Does consumer 1 prefer A over B? Does consumer 2 prefer C over A?
Xa = Xb and Yb > Ya, consumer 1 is able to trade in fewer units of x for more units of y
As a result consumer 1 refers B over A, by the monotonicity of preferences, which states if
A = (Xa, Ya) is a bundle of goods and B = (Xb, Yb) is a bundle of goods with at least as mush as one good or both good, in this case Yb, then B = (Xb, Yb) > A = (Xa, Ya).
Consider an economy with two (types of) consumers and two goods x and y.
Consumer 1 has endowment
ω1 = (8, 2) and consumer 2’s endowment is ω2 = (0, 4).
Suppose that consumer 1’s preferences are described by
U1(X1, Y1) = ln(X1) + ln(Y1)
and consumer 2’s preferences are described by
U2(X2, Y2) = ln(X2) + ln(Y2)
(a) Determine the set of Pareto efficient allocations.
- Find the marginal rate of substitution of both consumers
Consumer 1:
U1(X1, Y1) = ln(X1) + ln(Y1)
MRSxy(X1, Y1) = MUX/MUY
= 1/x1/ 1/y1 = y1/x1
Consumer 2:
U2(X2, Y2) = ln(X2) + ln(Y2)
MRSxy(X2, Y2) = MUX/MUY
= 1/x2/ 1/y2 = y2/x2
- Set MRS1xy = MRS2xy
y1/x1 = y2/x2 - Write the feasibility constraints with the endowments
x1 + x2 = 8
y1+ y2 = 6
- Substitute and solve for the set
y1/x1 = y2/x2
Plug the constraints: (Rearrange)
y1/x1 = 6-y1/8-x1
Solve for Y1
(8-x1)y1 = (6-y1)x1
y1(x1) = 3/4x1
Consider an economy with two (types of) consumers and two goods x and y.
Consumer 1 has endowment
ω1 = (8, 2) and consumer 2’s endowment is ω2 = (0, 4).
Suppose that consumer 1’s preferences are described by
U1(X1, Y1) = ln(X1) + ln(Y1)
and consumer 2’s preferences are described by
U2(X2, Y2) = ln(X2) + ln(Y2)
Find a competitive equilibrium.
- Find the Budget Constraints
Consumer 1:
Px1 + y1 = Pwa^x + wa^y
Px1 + y1 = 8P + 2
Consumer 2:
Px2 + y2 = Pwb^x + wb^y
Px2 + y2 = 0 + 4 = 4
- Maximize each consumers utility Let Px/Py = 1
Consumer 1:
MRSXY = px/py
MUX/MUY = px/py
1/x1 / 1/y1 = p / 1
Solve for Y1
Y1 = Px1
Consumer 2:
MRSXY = px/py
MUX/MUY = px/py
1/x2 / 1/y2 = p/1
Solve for Y2
Y2 = Px2
- Use the constraints
Consumer 1: Px1 + y1 = 8P + 2
Sub Y1 = Px1
Px1 + y1 = 8P + 2
Px1 + Px1 = 8P + 2
Solve for X1
x1 = 4 + 1/P
Consumer 2:
Px2 + y2 = 4
Sub Y2 = Px2
Px2 + Px2 = 4
Solve for X2
X2 = 2/P
- Check if the market clears
X1 + X2 = 8
4 + 1/P + 2/P = 8
Solve for P
P = 3/4
- Plug P = 3/4 into demand equations
X1 = 4 + 1/P = 16/3
X2 = 2/P = 8/3
Y1 = Px1 = 4
Y2 = Px2 = 2 - Check if the market clears for Y
Y1 + Y2 = 6
4 + 2 = 6
6 = 6
C) Now suppose next that consumer 1’s preferences are described by
U1(X1) = X1,
while consumer 2’s preferences are described by
U2(Y2) = 7Y2.
Find the set of Pareto effcient allocations in this case
Endowments
ω1 = (8, 2) and consumer 2’s endowment is ω2 = (0, 4).
- MRSXY^1 = MRSXY^2
Consumer 1:
Finding MRSXY^1 = MUX/MUY = 1/0 = 0
Consumer 2:
Finding MRSXY^2 = MUX/MUY = 0/7
MRSXY^1 = MRSXY^2
0 = 0
Consumer 1’s marginal utility for good y is zero and consumer 2’s marginal utility of good x is 0. This means allocating any amount of good x to consumer 2 or any amount of y to consumer 1 cannot be Pareto efficient, meaning it is possible to make one person better off without making the other person worse off. We could transfer whatever x and y they have to the other consumer to achieve Pareto improvement. The only Pareto efficient allocation is (8,6)
Now suppose next that consumer 1’s preferences are described by
U1(X1) = X1,
while consumer 2’s preferences are described by
U2(Y2) = 7Y2.
Does a competitive equilibrium exist for the consumers’ preferences described in (c)?
Endowments
ω1 = (8, 2) and consumer 2’s endowment is ω2 = (0, 4).
- Find the budget constraints
Consumer 1:
Px1 + y1 = 8P + 2
Consumer 2:
Px2 + y2 = 4
- Plug to constraints
Consumer 1:
y1 = 0 for consumer 1
Px1 + y1 = 8P + 2
Px1 + 0 = 8P + 2
Solve for X1
X1 = 8 + 2/P and Y1 = 0
Consumer 2:
x2 = 0 for consumer 2
Px2 + y2 = 4
0 + Y2 = 4
Y2 = 4
X2 = 0 and Y2 = 4
- Check to see if the market clears
The market for x
x1 + x2 = 8
8 + 2/P + 0 > 8
Check to see the market for y clears
y1 + y2 = 6
4 + 0 = 6
4 < 6
Therefore, there doesn’t exist any relative price that clears both markets
